Mayer functions

In the graphical representation of the integral shown above, a line represents the Mayer function f r.p between two particles and j. The coordinates are represented by open circles that are labelled, unless it is integrated over the volume of the system, when the circle representing it is blackened and the label erased. The black circle in the above graph represents an integration over the coordinates of particle 3, and is not labelled. The coefficient of is the sum of tln-ee tenns represented graphically as... 

The principle ideas and main results of tlie theory at the level of the second virial coefficient are presented below. The Mayer/-function for the solute pair potential can be written as the sum of temis ... 

Weak electrolytes in which dimerization (as opposed to ion pairing) is the result of chemical bonding between oppositely charged ions have been studied using a sticky electrolyte model (SEM). In this model, a delta fiinction interaction is introduced in the Mayer/-fiinction for the oppositely charged ions at a distance L = a, where a is the hard sphere diameter. The delta fiinction mimics bonding and tire Mayer /-function... 

To discuss briefly the reformulation of the Ornstein-Zernike equation it is most convenient to consider the case of one associating site per molecule, M = 1. A more general derivation can be found, for example, in Ref. 104. The most important ingredient for the following derivation is the associative Mayer function. It characterizes the bonding effects and is... 

Separating out the volume element fceU dr of one lattice cell, V0 and introducing the cell average Mayer function, (f), defined as ... 

In conclusion, the repulsive interactions arise from both a screened coulomb repulsion between nuclei, and from the overlap of closed inner shells. The former interaction can be effectively described by a bare coulomb repulsion multiplied by a screening function. The Moliere function, Eq. (5), with an adjustable screening length provides an adequate representation for most situations. The latter interaction is well described by an exponential decay of the form of a Bom-Mayer function. Furthermore, due to the spherical nature of the closed atomic orbitals and the coulomb interaction, the repulsive forces can often be well described by pair-additive potentials. Both interactions may be combined either by using functions which reduce to each interaction in the correct limits, or by splining the two forms at an appropriate interatomic distance . 

Thermodynamic perturbation theory is used to expand the Boltzmann distribution in the dipolar interaction, keeping it exact in the magnetic anisotropy (see Section II.B.l). A convenient way of performing the expansion in powers of is to introduce the Mayer functions fj defined by 1 +fj = exp( cOy), which permits us to write the exponential in the Boltzmann factor as... 

As a major deficit, in both DH and MSA theory the Mayer functions fxfi = exp —f q>ap(r) — 1 are linearized in ft. This approximation becomes unreasonable at low T and near criticality. Pairing theories discussed in the next section try to remedy this deficit. Attempts were also made to solve the PB equation numerically without recourse to linearization [202-204]. Such PB theories were also applied in phase equilibrium calculations [204-206]. 

The proper treatment of ionic fluids at low T by appropriate pairing theories is a long-standing concern in standard ionic solution theory which, in the light of theories for ionic criticality, has received considerable new impetus. Pairing theories combine statistical-mechanical theory with a chemical model of ion pair association. The statistical-mechanical treatment is restricted to terms of the Mayer/-functions which are linear in / , while the higher terms are taken care by the mass action law... 

For freely jointed flexible chain, the intrinsic energy can be defined by the r-mer Mayer function of an ideal chain,... 

Figure 6.1 Mayer-Montroll expansion for the insertion probability p(0 X"). The notation here is fairly standard (see, for example, Hansen and McDonald, 1976 Andersen, 1977). The solid lines indicate factors of Mayer / functions introduced in Eq. (6.2) and are further discussed as Ursell functions beginning on p. 126. The inclusion-exclusion interpretation for hard-core cases is that the second term - assesses the m molecular volumes excluded to the...

The prescription for determining the functions is that an m-decomposable model should be correct if the system consisted only of m molecules. Formulae such as Eqs. (6.2) and (6.3) involve the Mayer / function that, for a pair decomposable case, is / j) = exp [—/3m( )(0, y)] — 1. A natural accommodation of nonpair-decomposable interactions in this case takes the goal of insuring that successive terms in a virial expansion are ordered by the density. This is the historical approach (Ursell, 1927), and is called an Ursell expansion. In this language, fa (j) is an Ursell function (Stell, 1964 Munster, 1969). Again the idea is to require that the desired m-body Ursell function makes the product of Eq. (6.2) correct if just m molecules are involved. Thus for the case that only two molecules are involved... 

Figure 6.3 Compare with Fig. 6.1. Here again the solid lines indicate factors of Mayer / functions as in Eq. (6.25) and further discussed as Ursell functions beginning on p. 126. The second term shown is / f(r )p6 r r)d r for a simple fluid. The shaded regions with m-l black disks and one white disk represent conditional densities (rj,..., r). That white disk, rightmost here, corresponds to the real atom positioned at r on which the averages are conditioned. The other white disk, leftmost here, corresponds to the test particle.

With this set-up the / ]) is analogous to a Mayer / function for a hard object and can play the same role of monitoring the description of packing effects in liquids. See Exercise 6.1, p. 125 for an example. 

Mayer function for interaction between particles 1 and j (-) Faraday constant (C moL )... 

Thus, expandingFg(l, 2) in Mayer/functions [/= 1 — exp(—m/AF)] and introducing the result into Eq. (6.2), we find the second virial coefficient in the following form ... 

Improvements upon the theories require more detailed treatment of intramolecular as well as intermolecular interactions. As we mentioned in the previous section, use of Mayer /-functions has been made by Yamakawa and others to take intramolecular excluded volume effects into consid ation. However, in their calculation, parts of macromolecules between two consecutive contact points of two molecules are replaced always by Gaussian-free chains. While this approximation may be correct for a small number of contact points for very long molecules, it certainly invalidates the /-function expansion itself for higher orders. On the other hand, our results in the previous section indicate clearly that collective interactions of two macromolecules play important roles in explaining the molecular weight dependence of Ag. 

In the physical picture ion-pairs are just consequences of large values of the Mayer /-functions that describe the ion distribution [22], The technical consequence, however, is a major complication of the theory the high-temperature approximations of the /-functions applied, e.g. in the mean spherical approximation (MSA) or the Percus-Yevick approximation (PY) [25], suffice in simple fluids but not in ionic systems. 

The Mayer /-function is defined as the difference between the Boltzmann factor for two monomers at distance r and that for the case of no interaction (or at infinite distance) ... 

At short distances, the energy U r) is large because of the hard-core repulsion, making the Mayer /-function negative. The probability of - ding monomers at these distances is significantly reduced relative to the... 

The excluded volume v is defined as minus the integral of the Mayer /-function over the whole space ... 

The Mayer/-function and its integration (shaded regions) to determine excluded... 

As temperature is lowered, the Mayer /-function increases in the region of the attractive well, reducing the excluded volume. 

In a typical case of the Mayer /-function with an attractive well, repul-sion dominates at higher temperatures and attraction dominates at lower temperatures. In athermal solvents with no attractive well there is no temperature dependence of the excluded volume. It is possible to have monomer-solvent attraction stronger than the monomer monomer attraction. In this case, there is a soft barrier in addition to the hard-core repulsion and the excluded volume v>b d decreases to the athermal value v = b d at high temperatures. 

All results for chain size are now written in terms of the excluded volume. To understand how the chain size changes with temperature, we simply need the temperature dependence of the excluded volume. There are two important parts of the Mayer /-function, from which the excluded volume is calculated [Eq. (3.7)]. The first part is the hard-core repulsion, encountered when two monomers try to overlap each other (monomer separation rhard-core repulsion, the interaction energy is enormous compared to the thermal energy, so the Mayer /-function for r < 6 is — I ... 

The second part is for monomer separations larger than their size (r > b), where the magnitude of the interaction potential is small compared to the thermal energy. In this regime, the exponential can be expanded and the Mayer /-function is approximated by the ratio of the interaction energy and the thermal energy ... 

Mayer /-function, [dimensionless], p. 99 cutoff function above the gel point, [dimensionless], p. 227 cutoff function below the gel point, [dimensionless], p. 227... 

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